Study guide: Introduction to finite element methods

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Terminology: natural and essential boundary conditions

$$ (u',v') + (bu',v) = (f,v) + u'(L)v(L) - u'(0)v(0)$$

Note: forgetting the boundary terms implies $ u'(L)=u'(0)=0 $ (unless prescribe a Dirichlet condition)

Conditions on $ u' $ are simply inserted in the variational form and called natural conditions

Conditions on $ u $ at $ x=0 $ requires modifying $ V $ (through $ \baspsi_i(0)=0 $) and are known as essential conditions

Lesson learned.

It is easy to forget the boundary term when integrating by parts. That mistake may prescribe a condition on $ u' $!